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                <h1 id="_1">随机变量及其分布</h1>
<h2 id="_2">随机变量</h2>
<p>定义：设随机试验的样本空间为<span class="arithmatex">\(S=\{e\}\)</span>.<strong>随机变量</strong><span class="arithmatex">\(X(e)\)</span>是样本空间<span class="arithmatex">\(S\)</span>上的实值单值函数。称<span class="arithmatex">\(X＝X(e)\)</span>为随机变量.</p>
<blockquote>
<p>随机变量本质上是映射</p>
</blockquote>
<h2 id="_3">离散型随机变量及其分布律</h2>
<p>可能取值是有限个或可列无限多个的随机变量叫<strong>离散型随机变量</strong>
设离散型随机变量X所有可能取的值为<span class="arithmatex">\(x_k(k=1,2,\cdots)\)</span>，X取各个可能值的概率，即事件<span class="arithmatex">\(\{X=x_k\}\)</span>的概率为<span class="arithmatex">\(P\{X=x_k\}=p_k,k=1,2,\cdots\)</span>。称该式为X的<strong>分布律</strong>，可用表格表示。
有概率的定义，<span class="arithmatex">\(p_k\)</span>满足如下两个条件：</p>
<ol>
<li><span class="arithmatex">\(p_k\geq 0,k=1,2,\cdots\)</span></li>
<li><span class="arithmatex">\(\sum_{k=1}^\infty p_k=1\)</span></li>
</ol>
<h3 id="0-1">（0-1）分布</h3>
<p>设随机变量X只可能取0与1两个值，它的分布律是</p>
<div class="arithmatex">\[P\{X=k\}=p^k(1-p)^{1-k},k=0,1(0&lt;p&lt;1)\]</div>
<p>则称X服从以p为参数的(0-1)分布或两点分布</p>
<h3 id="_4">伯努利试验、二项分布</h3>
<p>设试验E只有两个可能结果：<span class="arithmatex">\(A\)</span>及<span class="arithmatex">\(\bar A\)</span>，则称E为<strong>伯努利试验</strong>。设<span class="arithmatex">\(P(A)=p(0&lt;p&lt;1)\)</span>，此时<span class="arithmatex">\(P(\bar A)=1-p\)</span>。将E独立重复地进行n次，则称这一串重复的独立试验为<strong>n重伯努利试验</strong>。以随机变量X表示n重伯努利试验中事件A发生的次数，则</p>
<div class="arithmatex">\[P\{X=k\}=\binom{n}{k}p^k(1-p)^{n-k}\]</div>
<p>这被称为<strong>二项分布</strong>，记为<span class="arithmatex">\(X\sim b(n,p)\)</span>。</p>
<blockquote>
<p>二项分布是多次两点分布，两点分布是单次二项分布，也就是<span class="arithmatex">\(X\sim b(1,p)\)</span></p>
</blockquote>
<h3 id="_5">泊松分布</h3>
<p>设随机变量X所有可能取的值为<span class="arithmatex">\(0,1,2,\cdots\)</span>，而取各个值的概率为</p>
<div class="arithmatex">\[P\{X=k\}=\frac{\lambda^ke^{-\lambda}}{k!},k=0,1,2,\cdots\]</div>
<p>其中<span class="arithmatex">\(\lambda&gt;0\)</span>是常数，则称X服从参数为<span class="arithmatex">\(\lambda\)</span>的<strong>泊松分布</strong>，记为<span class="arithmatex">\(X\sim\pi(\lambda)\)</span></p>
<p><strong>泊松定理</strong>：设<span class="arithmatex">\(\lambda&gt;0\)</span>是一个常数，n是任意正整数，设<span class="arithmatex">\(np_n=\lambda\)</span>，则对于任意固定的非负整数k，有</p>
<div class="arithmatex">\[\lim_{n\to\infty}\binom{n}{k}p_n^k(1-p_n)^{n-k}=\frac{\lambda^ke^{-\lambda}}{k!}\]</div>
<blockquote>
<p>泊松分布是对二项分布的近似</p>
</blockquote>
<h2 id="_6">随机变量的分布函数</h2>
<p>定义：设X是一个随机变量，x是任意实数，函数<span class="arithmatex">\(F(x)=P\{X\leq x\},-\infty&lt;x&lt;\infty\)</span>称为X的<strong>分布函数</strong>。
对于任意实数<span class="arithmatex">\(x_1,x_2(x_1&lt;x_2)\)</span>，有</p>
<div class="arithmatex">\[P\{x_1&lt;X\leq x_2\}=P\{X\leq x_2\}-P\{X\leq x_2\}=F(x_2)-F(x_1)\]</div>
<p>分布函数F(x)具有以下的基本性质：</p>
<ol>
<li>F(x)是一个不减函数</li>
<li><span class="arithmatex">\(0\leq F(x)\leq 1\)</span>，且
<span class="arithmatex">\(F(-\infty)=\lim_{x\to-\infty}F(x)=0,F(\infty)=\lim_{x\to\infty}F(x)=1\)</span></li>
<li>F(x)是右连续的</li>
</ol>
<p>反之，可证具备上述性质的函数F(x)必是某个随机变量的分布函数</p>
<h2 id="_7">连续性随机变量及其概率密度</h2>
<blockquote>
<p>不可能事件一定概率为零，概率为零不一定是不可能事件</p>
</blockquote>
<p>如果对于随机变量X的分布函数F(x)，存在非负可积函数f(x)，使对于任意实数x有</p>
<div class="arithmatex">\[F(x)=\int_{-\infty}^xf(t)dt\]</div>
<p>则称X为<strong>连续性随机变量</strong>，f(x)称为X的<strong>概率密度函数</strong>，简称<strong>概率密度</strong>.
概率密度具有以下性质：</p>
<ol>
<li><span class="arithmatex">\(f(x)\ge0\)</span></li>
<li><span class="arithmatex">\(\int_{-\infty}^\infty f(x)dx=1\)</span></li>
<li><span class="arithmatex">\(\forall x_1,x_2,x_1\le x_2,有P\{x_1&lt; X\le x_2\}=F(x_2)-F(x_1)=\int_{x_1}^{x_2}f(x)dx\)</span></li>
<li>若f(x)在点x处连续，则有F'(x)=f(x)
反之，满足性质1,2的函数也有某一随机变量的分布函数与其对应</li>
</ol>
<h3 id="_8">均匀分布</h3>
<p>若连续性随机变量X具有概率密度</p>
<div class="arithmatex">\[f(x)=\left\{\begin{aligned}&amp;\frac{1}{b-a},&amp;&amp;a&lt;x&lt;b,&amp;\\&amp;0,&amp;&amp;其他,&amp;\end{aligned}\right.\]</div>
<p>则称X在区间(a,b)上服从<strong>均匀分布</strong>，记为X~U(a,b)
其分布函数为</p>
<div class="arithmatex">\[F(x)=\left\{\begin{aligned}&amp;0,&amp;&amp;x&lt;a,&amp;\\&amp;\frac{x-a}{b-a},&amp;&amp;a\le x&lt;b,&amp;\\&amp;1,&amp;&amp;x\ge b,&amp;\end{aligned}\right.\]</div>
<h3 id="_9">指数分布</h3>
<p>若连续性随机变量X的概率密度为</p>
<div class="arithmatex">\[f(x)=\left\{\begin{aligned}&amp;\frac{1}{\theta}e^{-x/\theta},&amp;&amp;x&gt;0,&amp;\\&amp;0,&amp;&amp;其他,&amp;\end{aligned}\right.\]</div>
<p>其中<span class="arithmatex">\(\theta&gt;0\)</span>为常数，则称X服从参数为<span class="arithmatex">\(\theta\)</span>的<strong>指数分布</strong>，记为<span class="arithmatex">\(X\sim E(\theta)\)</span>
其分布函数为</p>
<div class="arithmatex">\[F(x)=\left\{\begin{aligned}&amp;1-e^{-x/\theta},&amp;&amp;x&gt;0,&amp;\\&amp;0,&amp;&amp;其他,&amp;\end{aligned}\right.\]</div>
<p>指数分布具有<strong>无记忆性</strong>：</p>
<div class="arithmatex">\[\forall s,t&gt;0,P\{X&gt;s+t|X&gt;s\}=P\{X&gt;t\}\]</div>
<blockquote>
<p>假如X表示元件寿命，元件使用了s小时，至少能再使用t小时的条件概率，和从开始使用算至少能用t小时的概率是相等的，元件对已使用事件没有记忆——这便是无记忆性</p>
</blockquote>
<h3 id="_10">正态分布</h3>
<p>若连续性随机变量X的概率密度为</p>
<div class="arithmatex">\[f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]</div>
<p>其中<span class="arithmatex">\(\mu,\sigma(\sigma&gt;0)\)</span>为常数，则称X服从参数为<span class="arithmatex">\(\mu,\sigma\)</span>的<strong>正态分布</strong>或<strong>高斯分布</strong>，记为<span class="arithmatex">\(X\sim N(\mu,\sigma^2)\)</span>
f(x)的图形具有以下性质：</p>
<ol>
<li>关于<span class="arithmatex">\(x=\mu\)</span>对称</li>
<li>当<span class="arithmatex">\(x=\mu\)</span>时取得最大值<span class="arithmatex">\(f(\mu)=\frac{1}{\sqrt{2\pi}\sigma}\)</span></li>
<li>在<span class="arithmatex">\(x=\mu\pm\sigma\)</span>有拐点，渐近线为x轴（<span class="arithmatex">\(\sigma\)</span>越小图形越尖）
其分布函数为</li>
</ol>
<div class="arithmatex">\[F(x)=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^x e^{-\frac{(t-\mu)^2}{2\sigma^2}}dt\]</div>
<p>特别地，当<span class="arithmatex">\(\mu=0,\sigma=1\)</span>时称随机变量X符合<strong>标准正态分布</strong>，其概率密度和分布函数分别用<span class="arithmatex">\(\varphi(x),\Phi(x)\)</span>表示；一般的正态分布可变换为标准正态分布——若随机变量<span class="arithmatex">\(X\sim N(\mu,\sigma^2)\)</span>，则<span class="arithmatex">\(Z=\frac{X-\mu}{\sigma}\sim N(0,1)\)</span></p>
<p>设<span class="arithmatex">\(X\sim N(0,1)\)</span>，若<span class="arithmatex">\(z_\alpha\)</span>满足条件<span class="arithmatex">\(P\{X&gt;z_\alpha\}=\alpha,0&lt;\alpha&lt;1\)</span>则称<span class="arithmatex">\(z_\alpha\)</span>为标准正态分布的<strong>上α分位数</strong></p>
<h2 id="_11">随机变量的函数的分布</h2>
<p>这一章研究对于已知的随机变量X，求Y＝g(X)的概率分布</p>
<p>对于离散型随机变量X，要求Y的分布律，只需依次求出g(X)，得到Y可能的取值，并把这些取值对应的X的概率加和即可</p>
<p>对于连续型随机变量X，要求Y的概率密度，要先写出Y的分布函数，通过解<span class="arithmatex">\(g(X)\leq y\)</span>得到X的不等式，然后用X的分布函数表达出Y的分布函数，求导即可得到Y的概率密度与X的概率密度的关系</p>
<p>例如：<span class="arithmatex">\(Y=X^2,F_Y(y)=P\{Y\leq y\}=P\{X^2\leq y\}=P\{-\sqrt y \leq X \leq \sqrt y\}=F_X(\sqrt y)-F_X(-\sqrt y)\)</span></p>
<blockquote>
<p>连续型随机变量的函数不一定是连续型随机变量</p>
</blockquote>
              
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